Finding Approximate Areas Under Curves

1. Finding Approximate Areas Under Curves

2. The Trapezium Rule y0 y1 y2 y3 y4 y5 This curve has a complicated equation so instead of integrating split the area up into a number of trapeziums each of width h and find the area of each. • y0, y1, y2, y3etc The y coordinates are given by

3. h y0 y1 y2 y3 y4 y5 Proving the Formula • Area of trapezium 1 = • Area of Trapezium 2 = • Area of Trapezium 3 = • Area of Trapezium 4 = • Area of Trapezium 5 = h= strip width = interval width • 1/2(y0 + y1) x h • 1/2(y1 + y2) x h • 1/2(y2 + y3) x h • 1/2(y3 + y4) x h • 1/2(y4 + y5) x h

4. h y0 y1 y2 y3 y4 y5 Proving the Formula • Area = 1/2h(y0 + y1+ y1+ y2 + y2…yn) • = 1/2h(y0 + 2y1+ 2y2 + 2y3…yn) • = 1/2h(y0 + 2(y1 + y2 + y3…) + yn)

5. Simpsons Rule Who was William Simpson?

6. Simpson was born in Market Bosworth, Leicestershire. The son of a weaver, Simpson taught himself mathematics, then turned to astrology after seeing a solar eclipse. He also dabbled in witchcraft and caused fits in a girl after 'raising a devil' from her. After this incident, he and his wife had to flee to London. From 1743, he taught mathematics at the Royal Military Academy, Woolwich. Apparently, the method that became known as Simpson's rule was well known and used earlier by Bonaventura Cavalieri (a student of Galileo) in 1639. It was later rediscovered by James Gregory (who Simpson succeeded as Professor of Mathematics at the University of St Andrews) but was only attributed to Simpson. In 1758, Simpson was elected a foreign member of the Royal Swedish Academy of Sciences. Simpson's rule is a staple of scientific data analysis and engineering. It is widely used, for example, by Naval architects to calculate the capacity of a ship or lifeboat.

7. y0 y1 y2 y3 y4 y5 Simpsons Rule This approximates the area under the curve using a quadratic curve to join the tops of each y value rather than a straight line as in the trapezium rule

8. (-h, y0), y0 (0, y1) y1 (h, y2) y2 0 h -h Consider three points which have coordinates (-h, y0), (0, y1), (h, y2). The tops of each red bar are joined by a quadratic curve(parabola) rather than straight lines as in the trapezium rule The parabola has the equation : y = ax2 + bx + c

9. y = ax2 + bx + c. y0 y1 y2 0 h -h The object is to find the area of the 2 strips shown using the y coordinates and then extend the idea to the next 2 strips.

10. y = ax2 + bx + c. y0 y1 y2 0 h -h A= Now substitute in the boundaries i.ex=h and x=-h

11. y = ax2 + bx + c. y0 y1 y2 0 h -h

12. y = ax2 + bx + c. y0 y1 y2 0 h -h A=

13. Remember the equation of the quadratic curve is y = ax2 + bx + c When we substitute the coordinates of the three points (-h, y0), (0, y1), (h, y2) into the equation for the parabola, we obtain the three equations y0 = ah2 - bh + c, y1 = c, y2 = ah2 + bh + c. If we add the 1st and 3rd equation then we obtain y0 + y2 = 2ah2 + 2c Now make a the subject

14. Remember the equation of the quadratic curve is y = ax2 + bx + c y0 = ah2 - bh + c, y1 = c, y2 = ah2 + bh + c. y0 + y2 = 2ah2 + 2c Now make a the subject

15. y1 = c Finally, substitute the above expression for a and y1 for c in the equation for the area: Cancel the h2 terms and the 2 h is a common factor

16. h is a common factor

17. h h h h h h

18. h h The formula works providing The number of y values is…………. And the number of strips is ……….. ODD EVEN

19. e.g. (a) Use Simpson’s rule with 4 strips to estimate giving your answer to 4 d.p. Solution: (a) y1 y0 y3 y4 y2

20. Solution:

21. (a) (b) Use your calculator to check the answer using the integration button

22. Exercise using Simpson’s rule 1. (a) Estimate with 2 strips, giving your answer to 4 d.p. (b) Improve your answer using 4 strips

23. 1. (a) Estimate using Simpson’s rule with 2 strips, giving your answer to 4 d.p. Solution: y0 y1 y2

24. 1. (a) Estimate using Simpson’s rule with 2 strips, giving your answer to 4 d.p. Solution:

25. 1. (a) Estimate using Simpson’s rule with 4 strips, giving your answer to 4 d.p. Solution: